Technically, the dog, while not captured, IS part of the captured gang fighter's equipment, and so is "in whatever lockbox" the captured gang fighter's equipment is in, while they are captured. The dog wouldn't go into the gang's stash any more than the captured gang fighter's pistol or goggles would go into the gang's stash while they are captured. So while they are captured, the dog wouldn't be transferred to another ganger.
That's not the awkward part.
The awkward part is that during most Rescue scenarios,* when deployed as a prisoner, the captured gang member doesn't get activation markers, but does immediately already have all of their equipment and weapons once freed. That would include the pet.
So the question is whether in most Rescue scenarios, the pet would be deployed with their owner (the prisoner). I mean, that COULD work, since pets don't get their own activation markers, but instead are part of a special group activation with their owners. Exotic Beasts can only be activated as part of a Group Activation with their owner. So if deployed next to their captured owner, in most Rescue scenarios, they could not be activated until their owner is freed. So they couldn't be used to help free their owner unfairly by starting next to their owner, past the sentries.
Alternately, phrasing could suggest that exotic pets are only "returned" to the captured-and-now-freed fighter when they are freed, which is too late to deploy their pet, so the pet has to sit that fight out (which mechanically makes some sense, but narratively is less fun); or that since exotic pets are then not "returned" to the fighter until they are freed, the exotic pet is suddenly deployed next to the freed fighter once they are freed (which seems odd to have them suddenly appear).
Narratively, it might seem fitting to have the trusted pet be part of the search for their captured owner -- but regardless, since the captive doesn't get activated in Rescue scenarios until they are freed, the Exotic Pet they own can also not get activated until the owner is freed, no matter where (or whether) that Pet could potentially be on the table.
